How to remove obstacles to learning math

I am adding a few comments about the article: Not a math person: "how to remove obstacles to learning math" written by Katrina Schwartz and published in the online magazine Mind/Shift concerning the pedagogical techniques to remove obstacles in learning math. The author of the article wrote: "Neuroscience research is now showing a strong connection between the attitudes and beliefs students hold about themselves and their academic performance". Our brain is the command center of our physical, emotional, mental and spiritual life. The brain stores also a lot of information recorded consciously or unconsciously. It interprets our surrounding reality and draws its conclusions. One type of conclusions is "the beliefs and attitudes" mentioned in the quote. Our beliefs and attitudes strongly influence our actions and personality. If students believe they can't do math it's obvious that they are not going to make any efforts in order to learn the subject. Their actions will reflect negative views or attitudes about learning math. The beliefs and attitudes originate from the student himself and his social and educational environment. If the social environment including the family cannot do much about developing positive attitudes about learning math this is the role of the school system to favor the development of attitudes and beliefs necessary for learning math. Not only the math teacher develops techniques to help students to learn the subject but he encourages students to "like" the subject. I use the word "like" instead of any other complicated word because I found when a student likes a subject he tends to make efforts in order to perform strongly at it.

"Neuroscientists now know that the brain has the abilities to grow and shrink". The fact that our brain grows simply means that we are able to use the brain to think, reflect and solve our problems. We know that every individual uses his brain to live. When we think for a certain period of time about something  we solicit the brain's resources in order to help us in order to solve problems. We use a great portion of the "working memory" of our brain to the solution of these problems. However we need to use the brain resources and the "working memory" effectively. Background knowledge helps us to free some parts of the working memory in order to move quickly in the solution of problems. In fact a study done about the strong performance of chess players shows that the memorization of different chess positions helps players to think quicker and gain advantages on their competitor. The background knowledge is important in performing math. Background knowledge is the knowledge of facts and theories so that we don't have to demonstrate them each time we encouter them. The fact that we know the multiplication tables allow us to do the multiplication of different large numbers instead of figuring out each time the multiplication of single numbers. There are math problems that are so complicated that we have to know the method of solutions and the formulas instead of figuring out each time how to solve the same type of problems. Background knowledge of math theories help us to solve complicated math problems. A complicated and abstract subject like math cannot be mastered without knowing its theories. The other technique mentioned in the article is about visualization. Visualization allows to visualize a fact, theory, etc. It helps to figure out something more clearly. It is a tool. It doesn't substitute the knowledge of the subject. Here is an excerpt of the article:   
Stanford math education professor Jo Boaler spends a lot of time worrying about how math education in the United States traumatizes kids. Recently, a colleague’s 7-year-old came home from school and announced he didn’t like math anymore. His mom asked why and he said, “math is too much answering and not enough learning.”

This story demonstrates how clearly kids understand that unlike their other courses, math is a performative subject, where their job is to come up with answers quickly. Boaler says that if this approach doesn’t change, the U.S. will always have weak math education.

“There’s a widespread myth that some people are math people and some people are not,” Boaler told a group of parents and educators gathered at the 2015 Innovative Learning Conference. “But it turns out there’s no such thing as a math brain.” Unfortunately, many parents, teachers and students believe this myth and it holds them up every day in their math learning.

There’s no such thing as a math brain.’Jo Boaler, Stanford professor of math education

“We live in a society with lots of kids who don’t believe they are good at math,” Boaler said at an Education Writers Association conference. “They’re put into low groups; they’re given low-level work and their pathway has been set.” But math education doesn’t have to look like this.

Neuroscience research is now showing a strong connection between the attitudes and beliefs students hold about themselves and their academic performance. That’s a departure from the long-held traditional view that academic success is based only on the quality of the teacher and curriculum. But researchers like Carol Dweck, Camille Farrington and David Yeager have shown repeatedly that small interventions to change attitudes about learning can have an outsized effect on performance.

Neuroscientists now know that the brain has the ability to grow and shrink. This was demonstrated in astudy of taxi drivers in London who must memorize all the streets and landmarks in downtown London to earn a license. On average it takes people 12 tries to pass the test. Researchers found that the hippocampus of drivers studying for the test grew tremendously. But when those drivers retired, the brain shrank. Before this, no one knew the brain could grow and shrink like that.

“We now know that when you make a mistake in math, your brain grows,” Boaler said. Neuroscientists did MRI scans of students taking math tests and saw that when a student made a mistake a synapse fired, even if the student wasn’t aware of the mistake. “Your brain grows when you make a mistake, even if you’re not aware of it, because it’s a time when your brain is struggling,” Boaler said. “It’s the most important time for our brains.”

A second synapse fires if the student recognizes his mistake. If that thought is revisited, the initial synapse firing can become a brain pathway, which is good for learning. If the thought isn’t revisited, that synapse will wash away.

A recent study of students with math learning disabilities found in a scan that their brains did behave differently from kids without the disability. “What they saw was the brain lighting up in lots of different areas while working on math,” Boaler said. The children were recruiting parts of the brain not normally involved in math reasoning.

The researchers tutored the group of students with math disabilities for eight weeks using the methods Boaler recommends like visualizing math, discussing problems and writing about math. At the end of the eight weeks, they scanned their brains again and found that the brains of the test group looked just like the kids who did not have math disabilities. This study shows that all kids can learn math when taught effectively. Boaler estimates that only 2 to 3 percent of people have such significant learning disabilities that they can’t learn math at the highest levels.

People who learned math the traditional way often push back against visual representations of math. That kind of thinking represents a deep misunderstanding of how the brain works. “When you think visually about anything, different brain pathways light up than when we think numerically,” Boaler said. The more brain pathways a student engages on the same problem, the stronger the learning.

An example of many ways to visually represent 18*5 (Jo Boaler/YouCubed) (to be continued)

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